DSCC2012-MOVIC

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1 ASME th Annual Dynamic Systems and Control Conference joint with the JSME th Motion and Vibration Conference DSCC2012-MOVIC2012 October 17-19, 2012, Fort Lauderdale, Florida, USA DSCC2012-MOVIC STORAGE FUNCTION FOR PASSIVITY ANALYSIS OF PNEUMATIC ACTUATORS WITH FINITE HEAT TRANSFER IN HUMAN-INTERACTIVE SYSTEMS Venkat Durbha Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota Perry Li Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota ABSTRACT The power density and variable compliance in pneumatic actuators makes them an attractive option for actuation in human assistive devices. Interaction safety in these devices can be robustly achieved through energetically passive controllers. Efficacy of these controllers depends on appropriate definition of actuator energy function. In previous works, the energy function was defined by assuming the thermodynamic process in the actuator to be either isothermal or adiabatic. In the current paper an estimate of work potential suitable for passivity analysis of a single chambered pneumatic actuator with finite heat transfer is reported. The energy function is developed by maximizing the work done on the actuator to reach an equilibrium position. Optimal conditions show that the maximal solution is attained if the thermodynamic process is a combination of adiabatic and isothermal processes. Through this storage function it is shown that the heat transfer has dissipative affect on the power flow in the pneumatic actuator, irrespective of the chamber air temperature. 1 INTRODUCTION Pneumatic actuators offer distinct advantages due their variable compliance, power density and clean work environment. These features make them an ideal actuator in human rehabilitative devices. Consequently pneumatic actuators are being explored for applications such as the orthotic device [1], the rescue crawler [2], and potentially as human power amplifiers [3], [4]. There is also on going research to improve energy density of pneumatic actuators through ideas such as the chemo-fluidic ac- Address all correspondence to this author. tuators [5], and small scale HCCI engines [6]. Higher energy density is achieved by using hot gases for providing actuation. These actuators can improve the viability of gas-based actuation in mobile human-scale applications. A key requirement in human interactive systems is interaction stability while exploring un-modeled environments. It has been shown in systems theory [7] that closed loop interaction between two stable systems is not always stable. As shown in [8], stability is guaranteed only if both the interacting systems are passive. Human muscle dynamics have been shown to behave passively in [9] and most external environments are passive in nature. Research effort on passivity framework for analysis and control of fluid powered systems is fairly recent. In [10], a novel approach to passive operation of hydraulic human power amplifier was reported. These ideas were extended to a pneumatic human power amplifier in [3], [11]. Energetic passivity analysis requires a model of the system energy function. In the previous work on pneumatic actuators, this was achieved by assuming the thermodynamic process in the actuator to be either isothermal or adiabatic. However the temperature change due to compression and expansion of gas in pneumatic systems results in finite heat interaction with the ambient. Due to the higher operating temperature in new actuation technologies being developed, the heat transfer will have significant effect on system dynamics. As explained in [12], a polytropic process is not accurate for modeling a thermodynamic process with finite heat transfer as it does not capture the energy loss due to hysteresis. In the current paper we report an energy based storage function to investigate passivity in single chamber pneumatic systems that has arbitrary finite heat interaction with a constant temperature ambient environment. The storage function is defined as the maximum work that can 1 Copyright 2012 by ASME

2 be extracted from the actuator for an optimal velocity and thermodynamic process. Passivity demonstrated with this storage function should hold for other possible thermal and mechanical interactions of the actuator. Optimality conditions demonstrate that the maximum work output is achieved if the thermodynamic process is a combination of adiabatic and isothermal processes. In [13], the work potential of compressed air is defined to be the exergy of air. Exergy as a measure of work potential has also been investigated for Lyapunov based controller design in [14]. The storage function presented in the current paper is shown to be the exergy of air undergoing a reversible thermodynamic process. Based on the storage function proposed in the current paper, heat transfer is shown to have dissipative effect for a general process irrespective of chamber temperature. In the following section the actuator dynamics are presented. The problem statement is presented in section 3. Design of the storage function is reported in section 4. The optimality of the proposed storage function is demonstrated by simulating a simple task in section 5. Concluding remarks are provided in section 6. 2 ACTUATOR DYNAMICS In the current work, for the sake of convenience a pneumatic actuator with a single air chamber, as shown in Fig. 1) is considered. It is assumed that air behaves as an ideal gas and therefore the pressure, temperature and volume are related as, P 1 V 1 = m 1 RT 1 1) where P 1 corresponds to the pressure, V 1 is the volume, m 1 is the mass and T 1 is the temperature of the air in actuator chamber. It is also assumed that primary mode of heat transfer is through conduction between the actuator chamber and the ambient. The pressure, temperature and heat transfer dynamics are given by [15], Ṗ 1 = ṁ1rt 1 + m 1RṪ 1 P 1 V 1 2) V 1 V 1 V 1 T1 Ṫ 1 = γt in T 1 )ṁ1 γ 1) V 1 T ) 1 Q 1 3) m 1 V 1 P 1 V 1 Q 1 = K h t,x)t o T 1 ) 4) where P 1 is the pressure, T 1 is the temperature, and m 1 is the mass of air in actuator chamber, γ is the ratio of specific heats, Q 1 is the heat transfer rate, T in is the temperature of air entering the chamber, T o is the ambient temperature, and K h is a positive quantity corresponding to product of effective heat transfer area and thermal conductivity of the material. By suitably defining K h, various models for conductive heat transfer can be captured. It is assumed that the effective heat transfer area can be changed as desired to simulate various possible heat transfer scenarios. The pressure and temperature dynamics can be changed by modulating the mass flow rate to the actuator chamber. The mass flow rate to the actuator chamber for choked and un-choked flow conditions is given by, ṁ 1 = Ψ 1 C f A v C 1 T P u Ψ 1 = P C u 2 T P d P u ) 1 γ 1 P d P u ) γ 1 γ if P d P u P cr choked flow) P d P u > P cr Un-choked) where P d corresponds to the pressure downstream of the valve, P u corresponds to the pressure upstream of the valve, and P cr = 0.58 is the critical pressure ratio differentiating the choked and unchoked flow regimes. The term C f A v represents the effective valve area. Modulation of the flow rate is achieved by changing the effective valve area and is therefore designated as the control input to achieve power amplification. 5) 3 PROBLEM STATEMENT A typical application for a human power amplification is as shown in Fig. 2). The dynamics of the inertia are given by, M p ẍ = F a + F h + F e 6) Figure 1. Schematic of a single chamber actuator studied in this paper where x is the position of the inertia, M p is the mass of the inertia, F a is the actuator force acting on the inertia, F h is the input human force and F e corresponds to other external forces acting on the system. The actuator force is given by, 2 Copyright 2012 by ASME

3 3) and Eq. 8), the heat transfer rate between the air chamber and the ambient, affects the actuator force. The energy transfer to/from the actuator due to heat transfer can affect the interaction dynamics of the actuator. These effects can be addressed by explicitly modeling the energy transferred through heat. An accurate model of heat transfer can get rather detailed [15]. A simpler approach would be to only monitor the direction of heat transfer and to define a storage function that would include the worst case scenario with heat transfer. Passive operation of human power amplifier demonstrated with this storage function will imply passive operation for all other possible scenarios of energy transfer through heat. To this end, the storage function is defined as the maximum work done by the actuator to reach equilibrium position, for a given air temperature, mass, and ambient temperature. The work done by the actuator is given by, Figure 2. Schematic of human power amplifier W = 0 m1 RT 1 L 1o + x P atma p u 3 dτ 12) F a = P 1 A 1 P atm A p 7) = m 1RT 1 L 1o + x P atma p 8) where A 1 and A p correspond to area on the cap side and area of the piston, and L 1o is the length corresponding to the dead volume. As the inertia is intended to move at the same velocity as the hand of the human operator, power amplification is achieved by scaling up the input human force. The desired dynamics of the system is given by, M L ẍ = ρ + 1)F h + F e 9) where u 3 = ẋ is the velocity of the actuator piston. The affect of external forces on the actuator power flow is captured through the velocity. Therefore, velocity is designated as a control variable for optimizing the work output. The effect of heat transfer on the work output is captured through the temperature dynamics. Different thermodynamic processes can be modeled by varying the heat transfer coefficient K h t). Therefore it is designated as another input to maximize the work output. The constraint that K h t) is positive is addressed by designating the corresponding input to be quadratic. For given mass, temperature and position, the ideal gas law in Eq. 1) defines the pressure. Therefore pressure dynamics are not explicitly included as a dynamic constraint. Therefore, the dynamic constraint imposed on the work output are given by, where M L = M p + M v ) is the inertia of the amplifier M v is the mass of a virtual inertia. See [10] for further details). The input human force is scaled by the factor ρ to define the desired actuator force. The energetic supply rate to the power amplifier is given by, ẋ = u 3 Ṫ 1 = γ 1) T 1u 3 L 1o + x + u2 1 T ) o T 1 ) m 1 R 13) sf h,f e,ẋ) = ρ + 1)F h + F e )ẋ 10) To maintain stability when interacting with unknown environments, the following inequality must be satisfied t, where u 2 1 is the coefficient of heat transfer rate denoted by K h in earlier section). The constraint imposed by the temperature dynamics is essentially the first law of thermodynamics. This can be seen by rewriting the above equation using the ideal gas law, t 0 sf h,f e,ẋ) c 2 o 11) where c 2 o represents the maximum energy that can be extracted from the amplifier by environment. As seen from Eq. 2), Eq. m 1 C v Ṫ 1 = P 1 A 1 ẋ + Q 1 14) The storage function is then obtained by solving the following optimization problem, 3 Copyright 2012 by ASME

4 The dynamics of the Lagrange multiplier are obtained as, L = max Wu 1,u 3,T 1,x) u 1,u 3 subject to Eq13) The supply rate of the power amplifier as determined using this storage function is required to satisfy the passivity condition in Eq. 11) for stable operation. 4 STORAGE FUNCTION As the dynamic constraints should be satisfied forall t, the following Hamiltonion is defined by augmenting the cost function, λ 1 = H T 1 = γ 1) λ 3 = H x = γ 1) m1 C v λ 1 u 3 + λ 1 u ) L 1o + x) m 1 C v m1 C v λ 1 L 1o + x) 2 T 1 u 3 22) From the terminal constraint on the lagrange multiplier we get, m1 RT 1 H = L 1o + x P atma p u 3 +λ 1 γ 1) T 1u 3 L 1o + x + u2 1 T o T 1 ) ) m 1 R + λ 3 u 3 15) where λ 1 and λ 3 are the Lagrange multipliers on the two dynamic constraints. The first order optimality conditions that determine maximum work output are given by, H u 1 = 0, From the above conditions we get, H u 3 = 0 16) λ 1 u 1 T o T 1 = 0 17) m1 RT 1 L 1o + x P T 1 atma p γ 1)λ 1 L 1o + x + λ 3 = 0 18) The second order optimality condition for maximum work output requires that the hessian matrix H u1 u 3 be negative definite. [ H u1 u 3 = 2 H Hu u 2 = 2 1 H u3 u 1 H u 2 3 H u1 u 3 ] < 0 19) As H u1 u 3 = H u3 u 1 = 0 and H u 2 3 = 0, a weaker semi-definite requirement is imposed on the hessian. This leads to the following requirement for a maximal solution, λ 1 T o T 1 ) < 0 20) lim λ 1t) = 0 t lim λ 3 t) = 0 23) t From Eq. 17) we have the following candidates for optimal solution, Lagrange optimal condition: λ 1 = 0 24) { Isothermal process if T 1 = T o Thermal optimal condition: u 1 = 0 if T 1 T o 25) Work can be extracted from the actuator until a thermodynamic dead state is reached. At the dead state there is no possibility of thermal or mechanical interaction between the actuator and the ambient. Therefore, the equilibrium position of the actuator is defined to satisfy the following condition, m 1 RT o L 1o + x P atma p = 0 26) where x is the equilibrium position. It can be noticed that if the chamber temperature is not T o, then the optimal solutions in Eq. 24) and Eq. 25) have to be traversed until the ambient temperature is attained to reach equilibrium position. 4.1 Lagrange Optimal Solution For the optimal solution λ 1 = 0, the hessian matrix is zero. Therefore, it is inconclusive if this optimal solution corresponds to a maximum or a minimum solution. As shown in the following proposition, for this solution, the available energy in the actuator remains constant. Proposition 1. The optimal solution λ 1 = 0 corresponds to no thermal and mechanical interaction with the ambient. 4 Copyright 2012 by ASME

5 Proof. When the optimal solution λ 1 = 0 is imposed, we also have λ 1 = 0. From Eq. 21) we therefore get, equilibrium. In such a case, for the adiabatic process we have T 1 > T o. From the definition of equilibrium position in Eq. 30) we get, u 3 = 0 27) From the definition of work in Eq. 12) we can infer that the above condition implies no work interaction. Using Eq. 27) in Eq. 22) we get, λ 3 = 0 28) On differentiating Eq. 18) and using Eq. 28) along with the optimality condition in λ 1 = 0, we get, x a > x i 31) Work done by the chamber during an isothermal process is given by [11], W iso = m 1 RT o log L1o + x i + P atm A p x i x) 32) L 1o + x and the work done by the chamber during adiabatic process is given by [11], u 1 = 0 29) The above condition implies that there is no thermal interaction between the actuator chamber and the ambient. As there is no heat or mechanical interaction between the actuator and the ambient, the energy in the actuator remains constant. 4.2 Isothermal Condition When the chamber temperature is T o, isothermal process satisfies the first order optimality condition. However, as seen from Eq. 20), the hessian is not negative semi-definite for this solution. Therefore optimality of this solution is also inconclusive. The other possible optimal solution u 1 = 0, represents an adiabatic process. In the following proposition, it is shown that work output for an isothermal process is greater than an adiabatic process. Proposition 2. Given that the chamber temperature is the ambient temperature T o, maximum work is extracted if the process is isothermal. Proof. Let x i represent the equilibrium position corresponding to the isothermal process and x a correspond to equilibrium position of the adiabatic process. From the definition of equilibrium position we have, m 1 RT o L 1o + x i P atm A p = 0 m 1 R T 1 L 1o + x a P atm A p = 0 30) where T 1 corresponds to the equilibrium temperature if the process were adiabatic. Let us first consider the case where the initial conditions are such that the chamber compresses to reach W adb = m 1 C v T o T 1 ) P atm A p x a x) 33) ) L1o + x γ 1 = m 1 C v T o 1 P atm A p x a x) 34) L 1o + x a Difference in the work done on the chamber for the two processes is given by, L1o + x W adb W iso = m 1 C v T o 1 L 1o + x a γ 1)log L1o + x i L 1o + x ) γ 1 Using Eq. 31) in the above equation we get, ) ) P atm A p x a x i ) 35) L1o + x γ 1 W adb W iso < m 1 C v T o 1 L 1o + x a L1o + x i γ 1)log ) 36) L 1o + x As the process is a compression, we have x > x a and x > x i. Therefore the right hand side of above equation is negative. This gives the following condition, W adb < W iso 37) Therefore the work done in an isothermal process is greater than adiabatic process. It can similarly be shown that for initial condition that require expansion to reach equilibrium, isothermal process produces more work output than an adiabatic process. 5 Copyright 2012 by ASME

6 4.3 Optimal Solution u 1 = 0 The optimal solution u 1 = 0 implies that the thermodynamic process in the actuator chamber is adiabatic. For an adiabatic process, from Eq. 21) we have, t1 t dλ t1 1 dx = γ 1) m 1 C v λ 1 t L 1o + x 38) where t is the current time and t 1 is some time in the future. On integrating both side of the above equation we get, Figure 3. Schematic showing work done when chamber temperature is greater than ambient temperature. Volume of chamber is plotted along the x-axis and pressure is plotted along the y-axis. λ 1 t) = m 1 C v 1 T ) 1t 1 ) + λ 1 t 1 ) T 1t 1 ) T 1 t) T 1 t) 39) From the equation for lagrange multiplier in Eq. 41) we obtain the following condition on temperature, where the following adiabatic relation is used to obtain the above equation, T 1 < T 1 t) 44) T 1 t)l 1o + xt)) γ 1 = T 1 t 1 )L 1o + xt 1 )) γ 1 40) From Eq. 23) and Eq. 39) we get the following equation for λ 1 t), From Eq. 40) we can see that above condition is achieved if the chamber expands. Therefore, if the chamber temperature is below the ambient, then the optimal process is adiabatic expansion. Similarly, it can be shown that if the chamber temperature is below the ambient temperature, the second order optimality condition is satisfied if the following condition is satisfied, λ 1 t) = m 1 C v 1 T ) 1 T 1 t) 41) T 1 > T 1 t) 45) where T 1 is the chamber temperature in the following limiting case for adiabatic process, T 1 = lim t T 1 t) 42) Proposition 3. If the chamber temperature is above ambient then the optimal process for extracting work is adiabatic expansion and if the chamber temperature is below ambient then the optimal process for extracting work is adiabatic compression. Proof. If the chamber temperature is different from ambient, adiabatic process satisfies the first order optimality condition in Eq. 16). To obtain a minimal solution, the second order optimality condition in Eq. 20) should also be satisfied. If the chamber temperature is above the ambient temperature, i.e T o T 1 ) < 0, then we get the following condition on lagrange multiplier, λ 1 > 0 43) From Eq. 40) it can be inferred that the optimal process would then be adiabatic compression. Given the definition of equilibrium state, the above proposition might seem trivial. However note that no assumptions about equilibrium state are made in obtaining the optimal process. The optimal process when the chamber temperature is below ambient is certainly not intuitive with out knowing the equilibrium position. Conclusions from proposition [2] and [3] are shown in Fig. 3) and Fig. 4). The schematic in Fig. 3), illustrates pressure Vs. volume plot for a scenario when the chamber temperature is above ambient. The volume V o is initial temperature and V 1 corresponds to a volume when the ambient temperature is attained. The red line in the plot corresponds to the initial adiabatic process. At ambient temperature, there are two options : isothermal process represented by the blue line and adiabatic process represented by the purple line. Work done during each process is given by the area under the graph. From the figure it is clear that work done by isothermal process is greater than adiabatic process. The schematic in Fig. 4) illustrates a scenario where the chamber temperature is lower than the ambient temperature. In 6 Copyright 2012 by ASME

7 W 1 = m 1 C v T 1 T o ) P atm A p x 1 x) 48) Figure 4. Schematic showing work done when chamber temperature is less than ambient temperature.volume of chamber is plotted along the x-axis and pressure is plotted along the y-axis. the figure, V o again corresponds to initial position and V 1 corresponds to the volume when chamber temperature is T o. the red line corresponds to the initial adiabatic process of compression to achieve ambient temperature. The blue line corresponds to the isothermal process. The purple line illustrates the solution corresponding to adiabatic process of expansion. From the plot it is evident that work done, as represented by the area under the graph, is greater if the process undergoes compression to reach ambient and then follows an isothermal trajectory. Note that the velocity of the actuator is not uniquely determined from the optimal solution. Therefore any velocity as determined by the interaction between the actuator and the environment is a feasible solution. The storage function for a single chamber pneumatic actuator is presented in the following section. where x 1 is the position of the actuator when the temperature is T o. As shown in proposition 2, when the temperature of the chamber corresponds to ambient temperature, isothermal process produces more work output than adiabatic process while attaining an equilibrium position. Maximal work for this phase of process is given by, L1o + x W 2 = m 1 RT o log P atm A p x x 1 ) 49) L 1o + x 1 where x is the equilibrium position. The current position x and x 1 are related as, We therefore have, T 1 L 1o + x) γ 1 = T o L 1o + x 1 ) γ 1 50) L1o + x L 1o + x 1 ) = L1o + x L 1o + x ) To T 1 ) 1 γ 1 51) 4.4 Exergy Theorem 1. For a pneumatic actuator with a single chamber, the storage function describing maximal work output is given by, T1 W pneu =m 1 C v T 1 T o ) m 1 T o C v log T o L1o + x + Rlog ) P atm A p x x) L 1o + x 46) The work done in isothermal phase can be written as, W 2 =m 1 RT o log L1o + x + 1 L 1o + x γ 1 log To T 1 )) P atm A p x x 1 ) 52) The expression for maximum available work output is given by, where x is the equilibrium position and is defined by, m 1 RT o L 1o + x P atma p = 0 47) Proof. Consider chamber temperature to be different from T o. From proposition 3 we know that the optimal solution is an adiabatic compression or expansion as determined by T o T 1 ). The maximum available work for this phase of the process is given by, W pneu = W 1 +W 2 L1o + x = m 1 C v T 1 T o ) m 1 T o Rlog L 1o + x T1 +C v log ) P atm A p x x) 53) T o Notice that for an initial temperature corresponding to T o, the storage function corresponds to that of an isothermal process. This has been shown to be a maximal solution in proposition 2. 7 Copyright 2012 by ASME

8 where γ 3 is obtained as, γ 3 = 1 W pneu Ψ 1 F a m 57) Figure 5. Schematic of the phase plot A phase plot of the optimal solution is as shown in Fig. 5). The x-axis is the actuator position and the y-axis is the chamber temperature. The origin corresponds to the equilibrium position, and ambient temperature. If the temperature of chamber is below the ambient, then the optimal solution is to compress the gas to reach ambient and then follow an isothermal path to equilibrium position. Similarly, when the temperature of the chamber air is greater than ambient, then the optimal path is to undergo expansion to reach ambient temperature and then reach equilibrium position through isothermal process. Let V 1 be the volume of the chamber at the equilibrium position of the actuator. The storage function can be written as, W pneu = m 1 C v T 1 T o ) m 1 T o s 1 s 1 ) P atm A p x x) 54) where s 1 is the specific entropy of air at temperature T 1 and volume V 1, and s 1 is similarly defined at equilibrium position. The above equation is obtained by using the following relation for a reversible process, V1 s 1 s 1 = Rlog +C v log V 1 T1 T o ) 55) By definition, the equilibrium position corresponds to a thermodynamic dead state of the actuator. Therefore the storage function in Eq. 54) represents the exergy of the air. Since the storage function is obtained through a combination of reversible processes, the relation between exergy and storage function intuitively makes sense. Using Eq. 4) in Eq. 56), we can infer that irrespective of the actuator chamber temperature, heat transfer has a dissipative effect on the power flow from the actuator. The actuator itself can be seen as a three port system, with one port interacting with the mechanical system, another port interacting with the fluid system with a pseudo velocity variable γ 3 u and the third port providing the thermal interaction. Due to its dissipative nature, any thermal interaction with an ambient at an arbitrarily fixed temperature with be passive and will only aid in stabilizing the system. Note that appropriate controllers still need to be designed to achieve human power amplification. As the heat transfer is not explicitly modeled, it is treated as an un-modeled disturbance in controller design. Controllers designed under the assumption of isothermal or adiabatic processes [11] can be extended using techniques such as nonlinear damping to achieve robust force amplification. The controller design and passivity analysis of the controller with respect to the storage function in Eq. 53) are not reported in this paper to keep the focus of the paper on the design of the storage function. 5 SIMULATION A simulation study was done to illustrate the conclusions of theorem 1. To this end, the storage function defined in Eq. 53) was compared with that of an adiabatic process as given by Eq. 34). The pressure and temperature dynamics were simulated by including heat transfer between the actuator and the ambient. The command input to the valve was designed such that the actuator follows a sinusoidal profile. While the the general trajectory of human power amplifier is arbitrary and task dependent, a sinusoidal profile was selected for convenience. Variation in the temperature as the actuator moves along a sinusoidal path is as shown in Fig. 6). The variation in optimized storage function and the adiabatic storage function is shown in Fig. 7). Both the storage functions are positive as desired. It is also evident from this plot that the available energy represented by the optimized storage function is more than the available energy represented by adiabatic non-optimized) process. Therefore, a controller shown to be passive with the optimized storage function should also be passive for other assumed thermodynamic processes. 4.5 Power Flow Analysis Differentiating the storage function in Eq. 53) we get, Ẇ pneu = F a ẋ + γ 3 F a u + Q 1 1 T ) o T 1 56) 6 CONCLUSION In this paper, storage function for a single-chamber pneumatic actuator with finite heat transfer with the ambient has been presented. It is shown that the reported storage function is similar to exergy of air for a reversible thermodynamic process. It has 8 Copyright 2012 by ASME

9 Temperature K) Figure 6. task Time s) Storage function Nm) Figure 7. Schematic showing the temperature variation in a simulated Non optimized model Optimized model Time s) Schematic comparing the optimized storage function with a non-optimized adiabatic) storage function also been shown that irrespective of actuator chamber temperature, heat transfer between the actuator and ambient at an arbitrarily fixed temperature has a dissipative affect on the available energy in the actuator. This seems intuitively accurate, as heat transfer is an irreversible process that results in entropy generation, which is quantified by destruction of useful available work. Extension of these results to double chambered actuator is currently under investigation. tion, 61), p. 19. [2] Guerriero, B., and Book, W., Haptic Feedback Applied to Pneumatic Walking. In ASME Dynamic Systems and Controls Conference. [3] P.Y.Li, and V.Durbha, Passive control of fluid powered human power amplifiers. In Japan Fluid Power Symp., Int. Fluid Power Symp., pp.. [4] Kazerooni, H., and Guo, J., Human extenders. Journal of Dynamic Systems, Measurement, and Control, 115, p [5] Fite, K., Mitchell, J., Barth, E., and Goldfarb, M., A unified force controller for a proportional-injector directinjection monopropellant-powered actuator. Journal of Dynamic Systems, Measurement, and Control, 128, p [6] Tian, L., Kittelson, D., and Durfee, W., Miniature HCCI Free-Piston Engine Compressor For Orthosis Application. SAE papers. [7] H.Khalil, Nonlinear Systems. Prentice Hall. [8] Willems, J., Dissipative dynamical systems, part 1: General theory. Arch. Rational Mech. Anal., 4522), pp [9] Hogan, N., Controlling impedance at the man/machine interface. Proc. IEEE Int. Conf. on Robotics and Auto., 1989., pp [10] Li, P., A new passive controller for a hydraulic human power amplifier. In IMECE, ASME, pp [11] Durbha, V., and Li, P., Passive tele-operation of pneumatic powered robotic rescue crawler. In FPNI Symposium. [12] Paynter, H., Cambridge, M., Fahrenthold, E., and Austin, T., On the Nonexistence of Simple Polytropes and Other Thermodynamic Consequences of the Dispersion Relation. Network Thermodynamics, Heat and Mass Transfer in Biotechnology: Presented at the Winter Annual Meeting of the American Society of Mechanical Engineers, Boston, Massachusetts, December 13-18, 1987, p. 37. [13] Kagawa, T., Cai, M., and Kawashima, K., Energy assessment in pneumatic systems and air power meter. In Bath Workshop On Power Transmission and Motion Control, Professional Engineering Publishing, pp [14] Robinett III, R., and Wilson, D., Exergy and irreversible entropy production thermodynamic concepts for nonlinear control design. International Journal of Exergy, 63), pp [15] Carneiro, J. F., and de Almeida, F., Reducedorder thermodynamic models for servo-pneumatic actuator chambers. In IMECE J. Sys. Control Engg., ASME, pp REFERENCES [1] Chin, R., Hsiao-Wecksler, E., Loth, E., Kogler, G., Manwaring, S., Tyson, S., Shorter, K., and Gilmer, J., A pneumatic power harvesting ankle-foot orthosis to prevent foot-drop. Journal of NeuroEngineering and Rehabilita- 9 Copyright 2012 by ASME